With the graph constructed we can now turn our attention to the
algorithm we will use to find the shortest solution to the word ladder
problem. The graph algorithm we are going to use is called the “breadth
first search” algorithm. Breadth first search (BFS) is one of
the easiest algorithms for searching a graph. It also serves as a
prototype for several other important graph algorithms that we will
study later.

Given a graph \(G\) and a starting vertex \(s\), a breadth
first search proceeds by exploring edges in the graph to find all the
vertices in \(G\) for which there is a path from \(s\). The
remarkable thing about a breadth first search is that it finds all the
vertices that are a distance \(k\) from \(s\) before it
finds any vertices that are a distance \(k+1\). One good way to
visualize what the breadth first search algorithm does is to imagine
that it is building a tree, one level of the tree at a time. A breadth
first search adds all children of the starting vertex before it begins
to discover any of the grandchildren.

To keep track of its progress, BFS colors each of the vertices white,
gray, or black. All the vertices are initialized to white when they are
constructed. A white vertex is an undiscovered vertex. When a vertex is
initially discovered it is colored gray, and when BFS has completely
explored a vertex it is colored black. This means that once a vertex is
colored black, it has no white vertices adjacent to it. A gray node, on
the other hand, may have some white vertices adjacent to it, indicating
that there are still additional vertices to explore.

The breadth first search algorithm shown in Listing 2 below uses the
adjacency list graph representation we developed earlier. In addition it uses a Queue,
a crucial point as we will see, to decide which vertex to explore next.

In addition the BFS algorithm uses an extended version of the Vertex
class. This new vertex class adds three new instance variables:
distance, predecessor, and color.

BFS begins at the starting vertex s and colors start gray to
show that it is currently being explored. Two other values, the distance
and the predecessor, are initialized to 0 and NULL respectively for
the starting vertex. Finally, start is placed on a Queue. The
next step is to begin to systematically explore vertices at the front of
the queue. We explore each new node at the front of the queue by
iterating over its adjacency list. As each node on the adjacency list is
examined its color is checked. If it is white, the vertex is unexplored,
and four things happen:

The new, unexplored vertex nbr, is colored gray.

The predecessor of nbr is set to the current node currentVert

The distance to nbr is set to the distance to currentVert+1

nbr is added to the end of a queue. Adding nbr to the end of
the queue effectively schedules this node for further exploration,
but not until all the other vertices on the adjacency list of
currentVert have been explored.

Let’s look at how the bfs function would construct the breadth first
tree corresponding to the graph in Figure 1. Starting
from fool we take all nodes that are adjacent to fool and add them to
the tree. The adjacent nodes include pool, foil, foul, and cool. Each of
these nodes are added to the queue of new nodes to expand.
Figure 3 shows the state of the in-progress tree along with the
queue after this step.

Figure 3: The First Step in the Breadth First Search

In the next step bfs removes the next node (pool) from the front of
the queue and repeats the process for all of its adjacent nodes.
However, when bfs examines the node cool, it finds that the color of
cool has already been changed to gray. This indicates that there is a
shorter path to cool and that cool is already on the queue for further
expansion. The only new node added to the queue while examining pool is
poll. The new state of the tree and queue is shown in Figure 4.

Figure 4: The Second Step in the Breadth First Search

The next vertex on the queue is foil. The only new node that foil can
add to the tree is fail. As bfs continues to process the queue,
neither of the next two nodes add anything new to the queue or the tree.
Figure 5 shows the tree and the queue after expanding all the
vertices on the second level of the tree.

Figure 5: Breadth First Search Tree After Completing One Level

FIgure 6: Final Breadth First Search Tree

You should continue to work through the algorithm on your own so that
you are comfortable with how it works. Figure 6 shows the
final breadth first search tree after all the vertices in
Figure 3 have been expanded. The amazing thing about the
breadth first search solution is that we have not only solved the
FOOL–SAGE problem we started out with, but we have solved many other
problems along the way. We can start at any vertex in the breadth first
search tree and follow the predecessor arrows back to the root to find
the shortest word ladder from any word back to fool. The function below (Listing 3) shows how to follow the predecessor links to
print out the word ladder.